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Sure, but you don't need to spend 32 weeks learning 10 different ways to do integrals which is what our calculuses classes would teach. Learn the concepts of derivatives and integrals and then move on to the actually interesting stuff you just mentioned here.

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You can't skip those steps without overwhelming your students. Advanced classes are already seen as hard even when students have practiced all this math a lot, remove the practice and you'll make the class impossible to pass for almost everyone.

I'll just continue to disagree, I guess. I was able to learn algebra just fine even though I didn't know my multiplication tables, while the rest of the class spent a year memorizing them. Our methods of deciding when someone "knows" something so they can "pass" needs to be radically rethought. We spend all our time practicing how to integrate by parts when in reality a computer will do that for you, and youre needed for higher level thinking and abstractions that the math classes don't bother to get to until graduate school. Sad imo.

There have been countless experiments along your lines of thinking, none of them has actually worked well at scale. If you think that educators love math, then think again, almost every teacher hated math when they were a student just like you do. And even though they hate it we still have it in its current form, that really tells you a lot about its merits.

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